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Operations on Functions




       Surgeons
For each of the following find         and their domains.




                                 Domain: 



                                 Domain: 


                                 Domain: 



                                  Domain: 
For each of the following find         and their domains.




                                 Domain: 



                                 Domain: 


                                 Domain: 



                                  Domain: 
For each of the following find         and their domains.




                                 Domain: 



                                 Domain: 


                                 Domain: 



                                  Domain: 
Composition of functions: 
 If f and g are functions, then their composite f  g is the
                                                  o

 function with 
                    [f   g](x) = f[g(x)]
                       o


 for each x in the domain of g such that g(x) is the 
 domain of f. 

 The function g is called the inner function, and its input is 
 x; f is the outer function and its input is g(x).
1
 If  f(x) = 2x + 1 and g(x) =  x

        find  [f   g](x)  and [g   f](x)
                 o               o




[f   g](x) 
   o                            Domain: 




                               Domain: 
[g   f](x)
   o
EVEN FUNCTIONS
Graphically: A function is "even" if its graph is symmetrical about the y­axis.


   These functions 
   are even...

                                                 These are 
                                                 not ...




Symbolically (Algebraically)
a function is "even" IFF (if and only if) ƒ(­x) = ƒ(x)
   Examples: Are these functions even?

  1. f(x) = x²                          2. g(x) = x² + 2x
      f(­x) = (­x)²                         g(­x) = (­x)² + 2(­x)
      f(­x) = x²                            g(­x) = x² ­ 2x
  since f(­x)=f(x)                      since g(­x) is not equal to g(x)
  f is an even function                 g is not an even function
ODD FUNCTIONS
Graphically: A function is "odd" if its graph is symmetrical about the origin.

    These 
    functions 
                                                These are 
    are odd ...                                 not ...




Symbolically (Algebraically)
a function is "odd" IFF (if and only if) ƒ(­x) = ­ƒ(x)
  Examples:       1. ƒ(x) = x³ ­ x                   2. g(x) = x³­ x²
                      ƒ(­x) = (­x)³ ­ (­x)               g(­x) = (­x)³ ­ (­x)²
                      ƒ(x) = ­x³ + x                     g(x) = ­x³ ­ x²

                      ­ƒ(x) = ­(x³ ­ x)                  ­g(x) = ­(x³­x²)
                      ­ƒ(x) = ­x³ + x                    ­g(x) = ­x³+ x²
                  since ƒ(­x)= ­ƒ(x)                 since g(­x) is not equal to ­g(x)
                  ƒ is an odd function               g is not an odd function
Exercise 1.9



Questions 1, 3, 5, 11, 15
Attachments



     Surgeons

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Comp Of Functions 0922